Network Model
Visualization

Understanding
complicated
network models
through visualization

Sam Tyner
@sctyner
@sctyner

Outline

Motivating Example

Childhood vaccination - major public health concern

Screenshot of animation from The Guardian

Questions

Interactions in animation aren’t realistic - people don’t behave like bacteria in petri dishes!

Can network modeling answer these?

Working Example

Another public health concern - underage drinking & smoking

How do friendships influence drinking and smoking behavior in teenagers?

Steglich, Snijders & Pearson (2010) “Dynamic Networks and Behavior: Searating Selection from Influence”

Data

From “Teenage Friends and Lifestyle Study” (Pearson & Mitchell (2000); Pearson & West (2003))

Model

Underlying Mechanisms?

Network changes

Rate Function

Rate at which actors are selected to act

Transition Steps

Objective Function

When actor \(i\) can change, it tries to maximize its objective function:

\[f_i(x, \mathbf{z}, \boldsymbol{\beta}) = \sum_k \beta_k s_{ik}(x, \mathbf{z})\]

Transition Probability

The probability that actor \(i\) will change the tie to actor \(j\) is:

\[p_{ij} = \frac{\exp\left\{f_i(x(i\leadsto j), \mathbf{z}, \boldsymbol{\beta})\right\}}{\sum_h \exp\left\{f_i(x(i\leadsto h), \mathbf{z}, \boldsymbol{\beta})\right\}}\] - \(x(i\leadsto j)\) is the network identical to the current state, \(x\), except for \(x_{ij}\), which becomes \(1-x_{ij}\)

Transition Probability Heatmap

(Some) Possible Model Parameters

Structural Effect Sufficient Statistic Interpretation
outdegree* \(s_{i1}(x) = \sum_j x_{ij}\) Popularity
reciprocity* \(s_{i2}(x) = \sum_j x_{ij}x_{ji}\) Reciprocated relationships
transitive triplets \(s_{i3}(x) = \sum_{j,h} x_{ij}x_{jh}x_{ih}\) Your friend becomes my friend
Covariate Effect Sufficient Statistic Interpretation
covariate-alter \(s_{i4}(x) = \sum_j x_{ij}z_j\) Effect of my friend’s behavior on friendship
covariate-ego \(s_{i5}(x) = z_i\sum_j x_{ij}\) Effect of my behavior on friendship
same covariate \(s_{i6}(x) = \sum_j x_{ij} \mathbb{I}(z_i = z_j)\) Birds of a feather flock together

(Some) Possible Model Parameters

Structural Effect Interpretation Visualization
outdegree* Popularity
reciprocity* Reciprocated relationships
transitive triplets Your friend becomes my friend
Covariate Effect Interpretation Visualization
covariate-alter Effect of my friend’s behavior on friendship
covariate-ego Effect of my behavior on friendship
same covariate Birds of a feather flock together

Small SAOMs Example

Static Visualizations

Distributions of Fitted Model Effects

Correlation of Estimates

Animation

GIF of Awesomeness

Interactivity with plotly

Future Work

References

Pearson, M. and Michell, L. 2000. “Smoke Rings: Social Network Analysis of Friendship Groups, Smoking, and Drug-Taking.” Drugs: Education, Prevention and Policy 7(1):21-37.

Pearson, M. and West, P. 2003. “Drifting Smoke Rings: Social Network Analysis and Markov Processes in a Longitudinal Study of Friendship Groups and Risk-Taking.” Connections 25(2):59-76.

Steglich, C., Snijders, T.A.B, and Pearson, M. 2010. “Dynamic Networks and Behavior: Separating Selection from Influence.” Sociological Methodology. 40(1):329-393.

Snijders, T.A.B. 1996. “Stochastic actor-oriented models for network change.” Journal of Mathematical Sociology 21:149-172.